An improved tableau criterion for Bruhat order

Anders Björner, Francesco Brenti

Abstract

To decide whether two permutations are comparable in Bruhat order of $S_n$ with the well-known tableau criterion requires $\binom{n}{2}$ comparisons of entries in certain sorted arrays. We show that to decide whether $x\le y$ only $d_1+d_2+...+d_k$ of these comparisons are needed, where $\{d_1,d_2,...,d_k\} = \{i|x(i)>x(i+1)\}$. This is obtained as a consequence of a sharper version of Deodhar's criterion, which is valid for all Coxeter groups.